Best Known (145, 145+39, s)-Nets in Base 4
(145, 145+39, 1056)-Net over F4 — Constructive and digital
Digital (145, 184, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 46, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
(145, 145+39, 4265)-Net over F4 — Digital
Digital (145, 184, 4265)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4184, 4265, F4, 39) (dual of [4265, 4081, 40]-code), using
- 154 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 40 times 0, 1, 65 times 0) [i] based on linear OA(4175, 4102, F4, 39) (dual of [4102, 3927, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(37) [i] based on
- linear OA(4175, 4096, F4, 39) (dual of [4096, 3921, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(38) ⊂ Ce(37) [i] based on
- 154 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 40 times 0, 1, 65 times 0) [i] based on linear OA(4175, 4102, F4, 39) (dual of [4102, 3927, 40]-code), using
(145, 145+39, 1662947)-Net in Base 4 — Upper bound on s
There is no (145, 184, 1662948)-net in base 4, because
- 1 times m-reduction [i] would yield (145, 183, 1662948)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 150 307280 082126 858286 748188 574530 137407 543039 573004 634338 825940 578418 265302 356068 135622 343612 474794 434431 698886 > 4183 [i]